Date: Feb 2, 2013 5:26 AM
Author: mina_world
Subject: Quadruple with (a, b, c, d)
Hello teacher~

Suppose not all four integers a, b, c, d are equal.

Start with (a, b, c, d) and repeatedly replace (a, b, c, d)

by (a-b, b-c, c-d, d-a).

Then at least one number of the quadruple will eventually

become arbitrarily large.

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Solution)

Let P_n = (a_n, b_n, c_n, d_n) be the quadruple after n iterations.

Then a_n + b_n + c_n + d_n = 0 for n >= 1.

A very importand function for the point P_n in 4-space

is the square of its distance from the origin (0,0,0,0),

which is (a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2.

....

omission

For reference, text copy with jpg.

http://board-2.blueweb.co.kr/user/math565/data/math/olilim.jpg

...

for n >=2,

(a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2

>= {2^(n-1)}*{(a_1)^2 + (b_1)^2 + (c_1)^2 + (d_1)^2}.

The distance of the points P_n from the origin increases without bound,

which means that at least one component must become arbitrarily large

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Hm, my question is...

I know that a_n + b_n + c_n + d_n = 0

and

I know that (a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2 increases without bound.

I can't understand that "at least one component must become arbitrarily

large".

I need your logical explanation or proof.